By Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin
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Additional info for Algebra and number theory : an integrated approach
It is important to note that the Euclidian Algorithm allows us to find these numbers x andy. Indeed, we have Thus so that rk = rk-2- (rk-3- rk-2qk-l)qk = rk-20 + qk-lqk)- rk-3qk = = rk-2- rk-3qk + rk-2qk-lqk rk-2YI- rk-3XJ, say. 4, we finally obtain the equation rk = ax + by. The values of x and y will then be evident. Of course many standard computer programs will compute the greatest common divisor of two integers in an instant. rk-3Y2 - rk-4X2. 1. Prove that 3 divides n 3 - n for each positive integer n.
Thus, if the associative law were to hold for commutation then it would follow that BAC +CAB= ACB + BCA. However, the matrices 50 ALGEBRA AND NUMBER THEORY: AN INTEGRATED APPROACH show that this is not true. In fact, BAC +CAB= c + (~ = = ACB+BCA = ~) (~ ~) (~ ~) ~) (~ ~) c~) (~ ~) (~ ~) + (~ nc ~) C~) + ( ~ ~) = G~) , (~ ~) G~) ~) c whereas c~) (~ ~) (~ ~) (~ ~) c ~) (~ ~) (~ ~) + = = + (~ ~) + (~ ~) = G~). Thus, BAC +CAB i= ACB + BCA, so we can see that [[A, B], C] i= [A, [B, C]]. However, for commutation there is a weakened form of associativity, known as the Jacobi identity which, states [[A, B], C] + [[C, A], B] + [[B, C], A]= 0.
Let a, b, c be integers. (i) If a divides be and a, b are relatively prime, then a divides c. (ii) If a, b are relatively prime and a, c are also relatively prime, then a and be are relatively prime. (iii) If a, b divide c and a, bare relatively prime, then ab divides c. Proof. 7 there are integers m, n such that am+ bn = 1. Then c = c(am = (ca)m + bn) = + (cb)n c(am) + c(bn) = (ac)m + (bc)n = a(cm) + (bc)n. 3(vi) shows that a divides a(cm) and hence a divides c. 7 there are integers m, n, k, t such that am+ bn ak + ct = 1.
Algebra and number theory : an integrated approach by Martyn R Dixon; Leonid A Kurdachenko; Igor Ya Subbotin